Looking at the controller, and learing a bit more about how MVC actually works, I was able to make sense of this.
My view was one of the auto-generated ones, and contained this line of code:
@Html.DropDownList("PriorityID", string.Empty)
To add html attributes, I needed to do something like this:
@Html.DropDownList("PriorityID", (IEnumerable<SelectListItem>)ViewBag.PriorityID, new { @class="dropdown" })
Thanks again to @Laurent for your help, I realise the question wasn’t as clear as it could have been…
UPDATE:
A better way of doing this would be to use DropDownListFor where possible, that way you don’t rely on a magic string for the name attribute
@Html.DropDownListFor(x => x.PriorityID, (IEnumerable<SelectListItem>)ViewBag.PriorityID, new { @class = "dropdown" })